Hybrid Gauss Pseudospectral and Generalized Polynomial Chaos Algorithm to Solve Stochastic Trajectory Optimization Problems

Abstract

A algorithm combining the Gauss Pseudospectral Method (GPM) with a Generalized Polynomial Chaos (gPC) method to solve nonlinear stochastic optimal control problems with constraint uncertainties is presented. The GPM and gPC have been shown to be spectrally accurate numerical methods for solving deterministic optimal control problems and stochastic differential equations, respectively. The gPC uses collocation nodes to sample the random space, which are then inserted into the differential equations and solved using standard solvers to generate a set of deterministic solutions used to characterize the distribution of the solution by constructing a polynomial representation of the output as a function of uncertain parameters. The proposed algorithm investigates using GPM optimization software in place of deterministic differential equation solvers traditionally used in the gPC, providing minimum cost deterministic solutions that meet path, control, and boundary constraints. A trajectory optimization problem is considered where the objectives are to find the path through a two-dimensional space that minimizes the probability a vehicle will be ’killed’ by lethal threats whose locations are uncertain and to characterize the effects those uncertainties have on the solution by estimating the statistical properties.